Syddansk Universitet
FDTD Simulation of Light Interaction with Cells for Nanobiophotonics Diagnostics and
Imaging
Tanev, Stoyan
Published in:
Handbook of Photonics for Biomedical Sciences
Publication date:
2010
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Tanev, S. (2010). FDTD Simulation of Light Interaction with Cells for Nanobiophotonics Diagnostics and
Imaging. In V. Tuchin (Ed.), Handbook of Photonics for Biomedical Sciences (pp. 3-45). C R C Press LLC.
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2
FDTD Simulation of Light Interaction with
Cells for Diagnostics and Imaging in
Nanobiophotonics
Stoyan Tanev
Integrative Innovation Management Unit, Department of Industrial and Civil Engineering, University of Southern Denmark, Niels Bohrs Alle 1, DK-5230 Odense M,
Denmark 1
Wenbo Sun
Science Systems and Applications, Inc., USA
James Pond
Lumerical Solutions, Vancouver, BC, Canada
Valery V. Tuchin
Research-Educational Institute of Optics and Biophotonics, Saratov State University,
Saratov, 410012, Russia, Institute of Precise Mechanics and Control of RAS, Saratov
410028, Russia
2.1
2.2
2.3
2.4
2.5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Formulation of the FDTD Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
FDTD Simulation Results of Light Scattering Patterns From Single Cells . .
FDTD Simulation Results of OPCM Nanobioimaging . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
5
25
30
39
39
41
This chapter describes the mathematical formulation of the Finite-Difference TimeDomain (FDTD) approach and provides examples of its applications to biomedical photonics problems. The applications focus on two different configurations light scattering from single biological cells and Optical Phase Contrast Microscope
(OPCM) imaging of cells containing gold nanoparticles. The validation of the FDTD
1 Formerly with the Technology Innovation Management Program in the Department of Systems and Com-
puter Engineering, Faculty of Engineering and Design, Carleton University, Ottawa, ON, Canada.
3
4
Handbook of Photonics for Biomedical Science
approach for the simulation of OPCM imaging opens a new application area with a
significant research potential — the design and modeling of advanced nanobioimaging instrumentation.
Key words: Finite-Difference Time-Domain (FDTD) method, light scattering, biological cell, gold nanoparticle, Optical Phase Contrast Microscope (OPCM) imaging,
optical clearing effect, image contrast enhancement, nanobiophotonics.
2.1 Introduction
The development of non-invasive optical methods for biomedical diagnostics requires a fundamental understanding of how light scatters from normal and pathological structures within biological tissue. It is important to understand the nature of the
light scattering mechanisms from micro-biological structures and how sensitive the
light scattering parameters are to the dynamic pathological changes of these structures. It is equally important to quantitatively relate these changes to corresponding
variations of the measured light scattering parameters. Unfortunately, the biological
origins of the differences in the light scattering patterns from normal and pathological (for example, pre-cancerous and cancerous) cells and tissues are not fully
understood. The major difficulty comes from the fact that most of the advanced optical biodiagnostics techniques have a resolution comparable to the dimensions of the
cellular and sub-cellular light scattering structures [1–2]. For example, conventional
Optical Coherence Tomography (OCT) techniques do not provide a resolution at the
sub-cellular level. Although there are already examples of ultrahigh resolution OCT
capabilities based on the application of ultra-short pulsed lasers [3–4], it will take
time beforesuch systems become easily commercially available. This makes the interpretation of images generated by typical OCT systems difficult and in many cases
inefficient. Confocal microscopy provides sub-cellular resolution and allows for diagnostics of nuclear and sub-nuclear level cell morphological features. However,
there are problems requiring a careful and not trivial interpretation of the images [5].
In its typical single photon form, confocal fluorescence microscopy involves an optical excitation of tissue leading to fluorescence that occurs along the exciting cone
of the focused light beam, thus increasing the chances of photo-bleaching of a large
area and making the interpretation of the image difficult.
In many nanobiophotonics diagnostics and imaging research studies optical software simulation and modeling tools provide the only means to a deeper understanding, or any understanding at all, of the underlying physical and biochemical processes. The tools and methods for the numerical modeling of light scattering from
single or multiple biological cells are of particular interest since they could provide
information about the fundamental light-cell interaction phenomena that is highly
relevant for the practical interpretation of cell images by pathologists. The compu-
FDTD Simulation of Light Interaction with Cells
5
tational modeling of light interaction with cells is usually approached from a single
particle electromagnetic wave scattering perspective which could be characterized by
two specific features. First, this is the fact that the wavelength of light is larger than
or comparable to the size of the scattering sub-cellular structures. Second, this is
the fact that biological cells have irregular shapes and inhomogeneous refractive index distributions which makes it impossible to use analytical modeling approaches.
Both features necessitate the use of numerical modeling approaches derived from
rigorous electromagnetic theory such as: the method of separation of variables, the
finite element method, the method of lines, the point matching method, the method of
moments, the discrete dipole approximation method, the null-field (extended boundary condition) method, the T-matrix electromagnetic scattering approach, the surface
Green’s function electromagnetic scattering approach, and the finite-difference time
domain (FDTD) method [6].
The FDTD simulation and modeling of the light interaction with single and multiple, normal and pathological biological cells and sub-cellular structures has attracted
the attention of researchers since 1996 [7–25]. The FDTD approach was first adopted
as a better alternative of Mie theory [26–27] allowing for the modeling of irregular
cell shapes and inhomogeneous distributions of complex refractive index values. The
emerging relevance of nanobiophotonics imaging research has established the FDTD
method as one of the powerful tools for studying the nature of light-cell interactions.
One could identify number of research directions based on the FDTD approach.
The first one focuses on studying the lateral light scattering patterns for the early
detection of pathological changes in cancerous cells such as increased nuclear size
and degrees of nuclear pleomorphism and nuclear-to-cytoplasmic ratios [7–17]. The
second research direction explores the application of FDTD-based approaches for
time-resolved diffused optical tomography studies [18–20]. A third direction is the
application of the FDTD method to the modeling of advanced cell imaging techniques within the context of a specific biodiagnostics device scenario [21–25]. An
emerging research direction consists in the extension of the FDTD approach to account for optical nanotherapeutic effects.
The present chapter will provide a number of examples illustrating the application
of the FDTD approach in situations associated with the first two research directions.
It is organized as follows. Section two provides a detailed summary of the formulation of the FDTD method including the basic numerical scheme, near-to-far field
transformation, advanced boundary conditions, and specifics related to its application
to the modeling of light scattering and optical phase contrast microscope (OPCM)
imaging experiments. Examples of FDTD modeling of light scattering from and
OPCM imaging of single biological cells are given in sections 3 and 4. The last
section summarizes the conclusions.
6
Handbook of Photonics for Biomedical Science
2.2 Formulation of the FDTD Method
2.2.1
The basic FDTD numerical scheme
The finite-difference time domain (FDTD) technique is an explicit numerical method
for solving Maxwell’s equations. It was invented by Yee in 1966 [28]. The advances
of the various FDTD approaches and applications have been periodically reviewed
by Taflove et al. [29]. The explicit finite-difference approximation of Maxwell’s
equations in space and time will be briefly summarized following Taflove et al. [29]
and Sun et al. [30–32].
In a source free absorptive dielectric medium Maxwell’s equations have the form:
∂H
,
∇ × E = −µ0
∂t
(2.1)
= ε0 ε ∂ E ,
(2.2)
∇×H
∂t
are the vectors of the electric and magnetic fields, respectively, µ0
where E and H
is the vacuum permeability and ε0 ε is the permittivity of the medium. Assuming a
harmonic [∝ exp(−iω t)] time dependence of the electric and magnetic fields and a
complex value of the relative permittivity ε = εr +iεi transforms Eq. (2.2) as follows:
∂ E
= ωε0 εi E + ε0 εr ∂ E ⇔ ∂ (exp(τ t)E) = exp(τ t) ∇ × H,
⇔ ∇×H
∂t
∂t
∂t
ε0 εr
(2.3)
where τ = ωεr /εi and ω is the angular frequency of the light. The continuous coordinates (x,y, z) are replaced by discrete spatial and temporal points: xi = i∆s, y j =
j∆s, zk = k∆s, tn = n∆t, where i = 0, 1, 2, · · · , I, j = 0, 1, 2, · · · , J, k = 0, 1, 2, · · · , K,
n = 0, 1, 2, · · · , N. ∆s and ∆t denote the cubic cell size and time increment, respectively. Using central difference approximations for the temporal derivatives over the
time interval [n∆t, (n + 1)∆t] leads to
= ε0 ε
∇×H
n+1/2 ,
E n+1 = exp(−τ ∆t)E n + exp(−τ ∆t/2) ∆t ∇ × H
(2.4)
ε0 εr
where the electric and the magnetic fields are calculated at alternating half-time steps.
The discretization of Eq. (2.1) over the time interval [(n − 1/2)∆t, (n + 1/2)∆t] (one
half time step earlier than the electric field) ensures second-order accuracy of the
numerical scheme. In a Cartesian coordinate system the numerical equations for the
x components of the electric and magnetic fields take the form
Hx
(i, j + 1/2, k + 1/2) = Hx
(i, j + 1/2, k + 1/2) + µ∆t
0 ∆s
×[Eyn (i, j + 1/2, k + 1) − Eyn (i, j + 1/2, k) + Ezn (i, j, k + 1/2) − Ezn (i, j + 1, k + 1/2)],
(2.5)
n+1/2
n−1/2
FDTD Simulation of Light Interaction with Cells
7
FIGURE 2.1: Positions of the electric- and the magnetic-field components in the
elementary (Yee) cubic cell of the FDTD lattice.
j,k)
n
Exn+1 (i + 1/2, j, k) = exp[− εεri (i+1/2,
(i+1/2, j,k) ω ∆t]Ex (i + 1/2, j, k)+
j,k)
exp[− εεri (i+1/2,
(i+1/2, j,k) ω ∆t/2] ε
∆t
0 εr (i+1/2, j,k)∆s
n+1/2
n+1/2
×[Hy
(i + 1/2, j, k − 1/2) − Hy
(i + 1/2, j, k + 1/2)
n+1/2
n+1/2
(i + 1/2, j + 1/2, k) − Hz
(i + 1/2, j − 1/2, k)],
+Hz
(2.6)
where Ex , Ey , Ez and Hx , Hy , Hz denote the electric and magnetic field components, respectively. The numerical stability of the FDTD scheme is ensured through
−1/2
,
the Courant-Friedrichs-Levy condition [29]: c∆t ≤ 1/∆x2 + 1/∆y2 + 1/∆z2
where c is the speed of light in the host medium and ∆x, ∆y, ∆y are the spatial
steps in the x, y and z direction, respectively. In our case ∆x = ∆y = ∆z = ∆s and
∆t = ∆s/2c. The positions of the magnetic and electric field components in a FDTD
cubic cell are shown in Fig. 2.1.
2.2.2
Numerical excitation of the input wave
We use the so-called total-field/scattered-field formulation [29, 32] to excite the
input magnetic and electric fields and simulate a linearly polarized plane wave propagating in a finite region of a homogeneous absorptive dielectric medium. In this
formulation, a closed surface is defined inside of the computational domain. Based
on the equivalence theorem [29], the input wave excitation within a given spatial domain can be replaced by the equivalent electric and magnetic currents located at the
8
Handbook of Photonics for Biomedical Science
FIGURE 2.2: Example of a closed rectangular surface separating the total fields
and scattered fields. The graph also shows the configuration of the one-dimensional
auxiliary FDTD gird that is used to calculate the input excitation fields [see the paragraph below Eq. (2.16)].
closed surface enclosing that domain. If there is a scatterer inside the closed surface,
the interior fields will be the total fields (incident plus scattered) and the exterior
fields are just the scattered fields. An example of geometrical configuration of such
closed surface (in this case rectangular) is shown in Fig. 2.2.
On the closed surface the electric ad magnetic field incident sources are added as
follows:
⇐H
− ∆t (Einc ×n),
H
µ0 ∆s
(2.7)
inc ),
E ⇐ E − ∆t (n × H
ε0 ε ∆s
(2.8)
inc are the incident fields and n is the inward normal vector of the
where Einc and H
closed surface [29]. Eqs. (2.5ab) take different forms at the different interfaces of
the closed rectangular surface O1 O2 O3 O4 O1 O2 O3 O4 shown in Fig. 2.2.
At the interface (i = ia , ..., ib ; j = ja − 1/2; k = ka + 1/2, ..., kb − 1/2):
n+1/2
Hx
n+1/2
(i, ja − 1/2, k) = {Hx
(i, ja − 1/2, k)}(1a) +
∆t n
E (i, ja , k);
µ0 ∆s z,inc
(2.9)
at the interface (i = ia , ..., ib ; j = jb + 1/2; k = ka + 1/2, ..., kb − 1/2):
n+1/2
Hx
n+1/2
(i, jb + 1/2, k) = {Hx
(i, jb + 1/2, k)}(1a) −
∆t n
E (i, jb , k); (2.10)
µ0 ∆s z,inc
FDTD Simulation of Light Interaction with Cells
9
at the interface (i = ia , ..., ib ; j = ja + 1/2, ..., jb − 1/2; k = ka − 1/2):
n+1/2
Hx
n+1/2
(i, j, ka − 1/2) = {Hx
(i, j, ka − 1/2)}(1a) −
∆t n
E (i, j, ka ); (2.11)
µ0 ∆s y,inc
at the interface (i = ia , ..., ib ; j = ja + 1/2, ..., jb − 1/2; k = kb + 1/2):
n+1/2
Hx
n+1/2
(i, j, kb + 1/2) = {Hx
(i, j, kb + 1/2)}(1a) +
∆t n
E (i, j, kb ); (2.12)
µ0 ∆s y,inc
at the interface (i = ia + 1/2, ..., ib − 1/2; j = ja ; k = ka , ..., kb ):
Exn+1 (i, ja , k) = {E n+1
j , k)}(1b)
x (i,
a
εi (i, ja ,k)
n+1/2
− exp − εr (i, j ,k) ω ∆t/2 ε εr (i,∆tj ,k)∆s Hz,inc (i, ja −1/2, k);
0
a
(2.13)
a
at the interface (i = ia +1/2, ..., ib −1/2; j = jb ; k = ka , ..., kb ):
Exn+1 (i, jb , k) = {E n+1
j , k)}(1b)
x (i,
b
εi (i, jb ,k)
n+1/2
+ exp − εr (i, j ,k) ω ∆t/2 ε εr (i,∆tj ,k)∆s Hz,inc (i, jb +1/2, k);
0
b
(2.14)
b
at the interface (i = ia + 1/2, ..., ib − 1/2; j = ja , ..., jb ; k = ka ):
Exn+1 (i, j, ka ) = {E n+1
j, k )}
x (i,
a (1b)
εi (i, j,ka )
n+1/2
+ exp − εr (i, j,k ) ω ∆t/2 ε εr (i,∆tj,k )∆s Hy,inc (i, j, ka −1/2);
a
0
(2.15)
a
at the face (i = ia + 1/2, ..., ib − 1/2; j = ja , ..., jb ; k = kb ):
j, k )}
Exn+1 (i, j, kb ) = {E n+1
x (i,
b (1b)
εi (i, j,kb )
n+1/2
− exp − εr (i, j,k ) ω ∆t/2 ε εr (i,∆tj,k )∆s Hy,inc (i, j, kb +1/2).
b
0
(2.16)
b
n+1/2
n+1/2
n , En
The incident fields Ey,inc
in Eqs. (2.9)–(2.12) and
z,inc and Hy,inc , Hz,inc
(2.13)–(2.16) are calculated by means of a linear interpolation of the fields on an
auxiliary one-dimensional linear FDTD grid. This auxiliary numerical scheme presimulates the propagation of an incident plane wave along a line starting at the origin
of the 3D grid m = 0, passing through the closest corner of the closed rectangular surface located at m = m0 , and stretching in the incident wave direction to a maximum
position m = mmax , as shown in Fig. 2.2. The incident wave vector kinc is oriented
with a zenith angle θ and an azimuth angle φ . mmax is chosen to be half of the total number of simulation time steps for the incident wave propagation in the entire
absorptive dielectric medium. Since it is impossible to use a transmitting boundary
condition for the truncation of the one-dimensional spatial domain, the selected mmax
value needs to ensure that no numerical reflection occurs at the forward end of the
one-dimensional grid before the 3D FDTD simulation ends. A Gaussian-pulse hard
wave source [29] is positioned at the m = 2 grid point in the form
10
Handbook of Photonics for Biomedical Science
2 t
n
−5
(m = 2) = exp −
.
(2.17)
Einc
30∆t
By using the hard wave source rather than a soft one at m = 2, the field at the grid
points m = 0 and 1 will not affect the field at the grid points m > 2. Therefore, there
is no need of boundary conditions at this end of the auxiliary one-dimensional FDTD
grid.
Assuming that a plane wave is incident from the coordinate origin to the closed
rectangular surface between the total- and scattered-fields as shown in Fig. 2.2, the
one-dimensional FDTD grid equations become
n+1/2
Hinc
n−1/2
(m + 1/2) = H inc
(m + 1/2)+
∆t
v (0,0)
µ0 ∆s[ v pp(θ ,φ ) ]
[E ninc (m) − E ninc (m + 1)],
(2.18)
n+1
n (m)
Einc
(m) = exp − εεri (m)
ω ∆t Einc
(m)
n+1/2
n+1/2
∆t
H
+ exp − εεri (m)
ω
∆t
(m
−
1/2)
−
H
(m
+
1/2)
,
v
(0,0)
inc
inc
p
(m)
ε0 εr (m)∆s
v p (θ ,φ )
(2.19)
where εi (m) and εr (m) denote the imaginary and real relative permittivity of the host
medium at position m, respectively. The equalization factor [v p (0, 0)/v p (θ , φ )] ≤ 1
is the numerical phase velocity ratio in the 3D FDTD grid [29].
2.2.3
Uni-axial perfectly matched layer absorbing boundary conditions
The FDTD numerical scheme presented here uses the Uni-axial Perfectly Matcher
Layer (UPML) suggested by Sacks et al. [33] to truncate the absorptive host medium
in the FDTD computational domain. The UPML approach is based on the physical introduction of absorbing anisotropic, perfectly matched medium layers at all
sides of the rectangular computational domain. The anisotropic medium of each
of these layers is uni-axial and is composed of both electric permittivity and magnetic permeability tensors. To match a UPML layer along a planar boundary to
a lossy isotropic half-space characterized by permittivity ε and conductivityσ , the
time-harmonic Maxwell’s equations can be written in forms [34–35].
y, z) = (iωε 0 ε + σ )sE(x, y, z),
∇×H(x,
(2.20)
y, z).
∇×E(x, y, z) = −iω µ 0 sH(x,
(2.21)
The diagonal tensor s is defined as follows
⎤⎡
⎤⎡
⎤ ⎡
⎤
0
sy 0 0
sy sz s−1
sz 0 0
s−1
x 0
x 0 0
⎦ ⎣ 0 sz 0 ⎦ = ⎣ 0
⎦ , (2.22)
sx sz s−1
s = ⎣ 0 sx 0 ⎦ ⎣ 0 s−1
y 0
y 0
−1
−1
0 0 sx
0 0 sz
0 0 sy
0
0
sx sy sz
⎡
FDTD Simulation of Light Interaction with Cells
11
σ
σz
σx
, sy = κy + iωεy 0 , and sz = κz + iωε
.
where sx = κx + iωε
0
0
The UPML parameters (κx , σx ), (κy , σy ), and (κz , σz ) are independent on the
medium permittivity ε and conductivity σ , and are assigned to the FDTD grid in the
UPML as follows: i) in the two absorbing layers at both ends of the computational
domain in the x-direction, σy = σz = 0 and κy = κz = 1; in the two layers at both
ends of the y-direction, σx = σz = 0 and κx = κz = 1; in the two layers at both ends
of the z−direction, σy = σx = 0 and κy = κx = 1; ii) at the x and y overlapping dihedral corners, σz = 0 and κz = 1; at the z and x overlapping dihedral corners, σy = 0
and κy = 1; iii) at all overlapping trihedral corners, the complete general tensor in
Eq. (2.22) is used. To reduce the numerical reflection from the UPML, several profiles have been suggested for incrementally increasing the values of (κx , σx ), (κy , σy )
and (κz , σz ). Here we use a polynomial grading of the UPML material parameters
[34–35]. For example,
κx (x) = 1 + (x/d)m (κx,max − 1),
(2.23)
σx (x) = (x/d)m σx,max ,
(2.24)
where x is the depth in the UPML and d is the UPML thickness in this direction.
The parameter m is a real number [35] between 2 and 4. κx,max and σx,max denote the
maximum κx and σx at the outmost layer of the UPML. For example, considering an
x-directed plane wave impinging at an angle θ upon a PEC-backed UPML with the
polynomial grading material properties, the reflection factor can be derived as [35]
R(θ ) = exp[−
2 cos θ
ε0 c
d
0
2σx,max d cos θ
σ (x)dx] = exp −
.
ε0 c(m + 1)
(2.25)
Therefore, if R(0) is the reflection factor at normal incidence, σx,max can be defined
as
σx,max = −
(m + 1) ln[R(0)]
.
2d/(ε0 c)
(2.26)
Typically, the values of R(0) are in the range between 10−12 to 10−5 and κx,max is
a real number between 1 to 30.
The UPML equations modify the FDTD numerical scheme presented by Eqs. (2.5)
and (2.6). The modified UPML FDTD numerical scheme is then applied to the entire
computational domain by considering the UPMLs as materials in a way no different
than any other material in the FDTD grid. However, this is not computationally
efficient. The usual approach is to apply the modified scheme only to the boundary
layers in order to reduce the memory and CPU time requirements. In the non-UPML
region, the unmodified FDTD formulation (Eqs. (2.5), (2.6)) is used. The derivation
of the modified UPML FDTD numerical scheme is not trivial at all. To explicitly
obtain the updating equations for the magnetic field in the UPML, an auxiliary vector
field variable B is introduced as follows [35]
12
Handbook of Photonics for Biomedical Science
Bx (x, y, z) = µ0 ( ssxz )Hx (x, y, z), By (x, y, z) = µ0 ( ssxy )Hy (x, y, z),
s
Bz (x, y, z) = µ0 ( syz )Hz (x, y, z).
Then Eq. (2.21) can be expressed as
⎡
⎤
∂ Ey (x,y,z)
⎤
⎡
⎤⎡
∂ Ez (x,y,z)
−
sy 0 0
Bx (x, y, z)
z
∂y
⎢ ∂ E ∂(x,y,z)
⎥
∂ Ex (x,y,z) ⎥
⎢ z
⎣
⎦ ⎣ By (x, y, z) ⎦ .
⎣ ∂ x − ∂ z ⎦ = iω 0 sz 0
∂
E
(x,y,z)
∂ Ex (x,y,z)
0 0 sx
Bz (x, y, z)
− y
∂y
(2.27)
(2.28)
∂x
On the other hand, inserting the definitions of sx , sy and sz into Eqs. (2.27) leads
to
(iωκx +
σx
σz
)Bx (x, y, z) = (iωκz + )µ0 Hx (x, y, z),
ε0
ε0
(2.29)
(iωκy +
σy
σx
)By (x, y, z) = (iωκx + )µ0 Hy (x, y, z),
ε0
ε0
(2.30)
(iωκz +
σy
σz
)Bz (x, y, z) = (iωκy + )µ0 Hz (x, y, z).
ε0
ε0
(2.31)
Now applying the inverse Fourier transform by using the identity iω f (ω ) →
∂ f (t)/∂ t to Eqs. (2.28) and (2.29)–(2.31) gives the equivalent time-domain differential equations, respectively
⎡
⎤
∂ Ey (x,y,z,t)
⎤
⎡
⎤⎡
∂ Ez (x,y,z,t)
−
κy 0 0
Bx (x, y, z,t)
∂z
∂y
⎢ ∂ E (x,y,z,t)
⎥
⎢ z
⎥ = ∂ ⎣ 0 κz 0 ⎦ ⎣ By (x, y, z,t) ⎦
− ∂ Ex (x,y,z,t)
⎣
⎦ ∂t
∂x
∂z
∂
E
(x,y,z,t)
∂ Ex (x,y,z,t)
y
0 0 κx
Bz (x, y, z,t)
−
(2.32)
∂x
⎤
⎡∂ y
⎤⎡
σy 0 0
Bx (x, y, z,t)
+ ε10 ⎣ 0 σz 0 ⎦ ⎣ By (x, y, z,t) ⎦ ,
0 0 σx
Bz (x, y, z,t)
κx
∂ Bx (x, y, z,t) σx
∂ Hx (x, y, z,t)
σz
+ Bx (x, y, z,t) = µκz
+ µ Hx (x, y, z,t),
∂t
ε0
∂t
ε0
(2.33)
κy
∂ By (x, y, z,t) σy
∂ Hy (x, y, z,t)
σx
+ By (x, y, z,t) = µκx
+ µ Hy (x, y, z,t),
∂t
ε0
∂t
ε0
(2.34)
κz
σy
∂ Bz (x, y, z,t) σz
∂ Hz (x, y, z,t)
+ Bz (x, y, z,t) = µκy
+ µ Hz (x, y, z,t).
∂t
ε0
∂t
ε0
(2.35)
After discretizing Eqs. (2.32) and (2.33)–(2.35), we can get the explicit FDTD
formulations for the magnetic field components in the UPML [29, 32]:
FDTD Simulation of Light Interaction with Cells
13
2ε κ −σ ∆t
n+1/2
n−1/2
Bx
(i, j + 1/2, k + 1/2) = 2ε 00 κyy +σ yy ∆t Bx
(i, j + 1/2, k + 1/2)
n
+ 2ε20εκ0y∆t/∆s
+σ y ∆t [Ey (i, j + 1/2, k + 1)
−Eyn (i, j + 1/2, k) + E nz (i, j, k + 1/2) − E nz (i, j + 1, k + 1/2)],
n+1/2
n−1/2
σ z ∆t
Hx
(i, j + 1/2, k + 1/2) = 22εε 00 κκzz −
(i, j + 1/2, k + 1/2)
Hx
+
σ
∆t
z
µ
[(2ε 0 κx +σ x ∆t)Bn+1/2
+ 2ε 0 κ1/
(i, j + 1/2, k + 1/2)
x
z +σ z ∆t
(2.36)
(2.37)
−(2ε0 κx −σ x ∆t)Bn−1/2
(i, j + 1/2, k + 1/2)].
x
are
Similarly, for electric field in the UPML, two auxiliary field variables P and Q
introduced as follows [32, 35]
sy s z
(2.38)
Px (x, y, z) =
Ex (x, y, z),
sx
sx s z
Py (x, y, z) =
(2.39)
Ey (x, y, z),
sy
sx s y
Pz (x, y, z) =
(2.40)
Ez (x, y, z),
sz
1
Qx (x, y, z) =
(2.41)
Px (x, y, z),
sy
1
Qy (x, y, z) =
(2.42)
Py (x, y, z),
sz
1
Qz (x, y, z) =
(2.43)
Pz (x, y, z).
sx
Inserting Eqs. (2.38)–(2.40) into Eq. (2.20), simply following the steps in deriving
Eq. (2.36), leads to the updating equations for the P components:
Pxn+1 (i + 1/2, j, k) =
2ε0 ε −σ ∆t
2ε0 ε +σ ∆t
n+1/2
Pxn (i + 1/2, j, k) +
2∆t/∆s
2ε0 ε +σ ∆t
n+1/2
(2.44)
×[Hy
(i + 1/2, j, k − 1/2) − Hy
(i + 1/2, j, k + 1/2)
n+1/2
n+1/2
(i + 1/2, j + 1/2, k) − Hz
(i + 1/2, j − 1/2, k)].
+Hz
From Eqs. (2.41)–(2.43), in an identical way to the derivation of Eq. (2.37), leads
components:
to the updating equations for the Q
×[Pxn+1 (i + 1/2,
2ε0 κy −σy ∆t
n
2ε0 κy +σy ∆t Qx (i + 1/2,
j, k) − Pxn (i + 1/2, j, k)].
Qn+1
x (i + 1/2, j, k) =
j, k) +
2ε 0
2ε0 κy +σy ∆t
(2.45)
14
Handbook of Photonics for Biomedical Science
Inserting Eqs. (2.38)–(2.40) into Eqs. (2.41)–(2.43) and also following the procedure in deriving Eq. (2.37), leads to the electric field components in the UPML:
Exn+1 (i + 1/2, j, k) =
2ε0 κz −σz ∆t
2ε0 κz +σz ∆t
×[(2ε0 κx + σx ∆t)Qn+1
x (i + 1/2,
2.2.4
Exn (i + 1/2, j, k) +
1
2ε0 κz +σz ∆t
n
j, k) − (2ε0 κx − σx ∆t)Qx (i + 1/2,
j, k)].
FDTD formulation of the light scattering properties from single
cells
The calculation of the light scattering and extinction cross sections by cells in free
space requires the far-field approximation for the electromagnetic fields [29, 36].
The far-field approach has been also used [37] to study scattering and absorption by
spherical particles in an absorptive host medium. However, when the host medium is
absorptive, the scattering and extinction rates depend on the distance from the cell.
Recently, the single-scattering properties of a sphere in an absorptive medium have
been derived using the electromagnetic fields on the surface of the scattering object
based on Mie theory [38–39]. Here we derive the absorption and extinction rates for
an arbitrarily-shaped object in an absorptive medium using the internal electric field
[32]. The absorption and extinction rates calculated in this way depend on the size,
shape and optical properties of the scattering object and the surrounding medium,
but do not depend on the distance from it. The singe particle scattering approach is
perfectly applicable to studying the light scattering properties from single biological
cells.
Amplitude scattering matrix
For electromagnetic waves with time dependence exp(−iω t)propagating in a chargefree dielectric medium, we can write Maxwell’s equations in the frequency domain
as follows
∇×D = 0,
= 0,
∇×H
∇×E → = iω µ 0 H,
= −iω D.
∇×H
(2.46)
The material properties of the host medium are defined by the background permittivity εh and the electric displacement vector is defined as
D = ε 0 εh E + P = ε 0 ε E,
(2.47)
where here P is the polarization vector. Given Eq. (2.47), the first and last equations
in Eqs. (2.46) lead to
∇ · E = −
1
∇ · P,
ε0 εh
= −iω (ε 0 εh E + P).
∇×H
(2.48)
(2.49)
FDTD Simulation of Light Interaction with Cells
15
Combining the third equation in Eqs. (2.46) and Eqs. (2.48), (2.49) yields a
source-dependent form of the electromagnetic wave equation
1 2
[k P + ∇(∇ · P)],
(∇ + kh2 )E = −
ε0 εh h
(2.50)
1
(k2 II + ∇∇) · P.
(∇ + kh2 )E = −
ε0 εh h
(2.51)
√
where kh = ω µ0 ε0 εh is the complex wave number in the host medium. Using the
unit dyad II=xx +yy +zz (wherex,y andz are unit vectors in the x, y and z direction,
respectively), we can rewrite Eq. (2.50) in the form
Eq. (2.47) leads to P = ε 0 (ε − ε h )E which means that P is nonzero only in the
region inside the cell. The general solution of Eq. (2.51) is given by a volume
integral equation [38]:
E(R) = E 0 (R)+
V
2
G(R, ξ )(kh II+∇ξ ∇ξ ) × (P/(ε 0 εh ))d 3ξ ,
(2.52)
where E0 (R) can be any mathematical solution of (∇2 +k2h )E = 0 but, in practice, the
only nontrivial solution here is the incident field in the host medium. The integration
volume v is the region inside the particle and G(R, ξ ) is the 3D Green function in the
host medium:
exp(ikh |R − ξ |)
G(R, ξ ) =
.
4π |R − ξ |
(2.53)
The scattered field in the far-field region can then be derived from Eq. (2.52) [36]:
Es (R)
ε (ξ )
− 1] E(ξ ) −r[r · E(ξ )] exp(−ikhr.ξ )d 3ξ .
εh
V
(2.54)
To calculate the amplitude scattering matrix elements, the incident and the scattered fields are decomposed into their components parallel and perpendicular to the
scattering plane (Fig. 2.3).
The incident field is decomposed in two components along the unit vectorseα and
eβ both laying in the X −Y plane and defined as parallel and perpendicular to the
scattering plane, respectively:
k2 exp(ikh R)
= h
4π R
kh R→∞
[
E0 =eα E0,α +eβ E0,β .
(2.55)
E0,α and E0,β are related to the x-polarized and y-polarized incident fields used in
the FDTD simulation with
E0,α
E0,y
β ·x −β ·y
= .
(2.56)
E0,β
E0,x
β ·y β ·x
16
Handbook of Photonics for Biomedical Science
FIGURE 2.3: Incident and scattering wave configurations. The incident wave is
propagating in the Z-direction. The unit vectors corresponding to the three coordi
nate axes are:
x,y,z. The scattering direction is defined by the vector R with a unit
vectorr = R /R. The z coordinate axis and the vector R define the scattering plane.
The unit vector α is in the scattering plane and is perpendicular to R andr. The unit
vector β is perpendicular to the scattering plane and β × α =r. The unit vectors eα
and eβ are in the X −Y plane and are, respectively, parallel and perpendicular to the
scattering plane. All vectors in the figure are in bold.
β :
The scattered field is decomposed in two components along the unit vectors α and
Es (R) = α Es,α (R) + β Es,β (R).
(2.57)
It is important to note that the incident and scattered fields are specified relative
to different sets of basis vectors. The relationship between the incident and scattered
fields can be conveniently written in the following matrix form
Es,α (R)
Es,β (R)
=
exp(ikh R) S2 S3
E0,α
,
E0,β
S4 S1
−ikh R
(2.58)
where S1 , S2 , S3 and S4 are the elements of the amplitude scattering matrix and, in
general, depend on the scattering angle θ and the azimuth angle φ .The combination
of Eqs. (2.54) and (2.58) leads to the following expressions for amplitude scattering
matrix:
FDTD Simulation of Light Interaction with Cells
Fα ,y Fα ,x
S2 S3
β ·x β ·y
.
S=
=
Fβ ,y Fβ ,x
S4 S1
−β ·y β ·x
17
(2.59)
The quantities Fα ,x , Fβ ,x and Fα ,y , Fβ ,y are calculated for x- and y-polarized incident light, respectively, as follows [36–38]:
for x-polarized incidence:
ikh3
ε (ξ )
α · E(ξ )
Fα ,x
ξ )d 3ξ ,
r
·
(2.60)
1−
=
exp(−ik
h
β · E(ξ )
Fβ ,x
4π V
εh
for y-polarized incidence:
ikh3
ε (ξ )
α · E(ξ )
Fα ,y
3
1−
=
β · E(ξ ) exp(−ikhr · ξ )d ξ ,
Fβ ,y
4π V
εh
(2.61)
√
where kh = ω µ0 ε0 εh and εh is the complex relative permittivity of the host medium.
When εh = 1, Eqs. (2.60), (2.61) will degenerate into a formulation for light scattering by cells in free space.
Eq. (2.58) is now fully defined and can be rewritten in a vectorial form:
Es = S · E0 ,
(2.62)
where Sk = Sk (θ , φ ), k = 1, 2, 3, 4. In actual experiments the measured optical signal is proportional to quadratic field combinations [40]. Therefore, to describe the
monochromatic transverse wave one introduces four Stokes parameters which in the
case of the scattered wave take the form [37]
Is = Es,α Es,∗ α + Es,β Es,∗ β ,
Qs = Es,α Es,∗ α − Es,β Es,∗ β ,
Us = Es,α Es,∗ β + Es,β Es,∗ α ,
(2.63)
Vs = Es,α Es,∗ β − Es,β Es,∗ α .
The relation between the incident and the scattered Stokes parameters is given by
the Mueller scattering matrix (or simply the scattering matrix) which is defined as
follows
⎛
⎞⎛ ⎞
⎛ ⎞
P11 P12 P13 P14
Ii
Is
⎜ P21 P22 P23 P24 ⎟ ⎜ Qi ⎟
⎜ Qs ⎟
1
⎜
⎟⎜ ⎟
⎜ ⎟=
(2.64)
⎝ Us ⎠ k2 R2 ⎝ P31 P32 P33 P34 ⎠ ⎝ Ui ⎠ ,
h
Vs
P41 P42 P43 P44
Vi
18
Handbook of Photonics for Biomedical Science
where
P11 =
1 2
|S1 | + |S2 |2 + |S3 |2 + |S4 |2 ,
2
P13 = Re(S2 S3∗ + S1 S4∗ ),
P21 =
1 2
|S2 | − |S1 |2 − |S4 |2 + |S3 |2 ,
2
P12 =
1 2
|S2 | − |S1 |2 + |S4 |2 − |S3 |2 ,
2
P14 = Im(S2 S3∗ − S1 S4∗ ),
P22 =
1 2
|S2 | + |S1 |2 − |S4 |2 − |S3 |2 ,
2
and
P23 = Re(S2 S3∗ − S1 S4∗ ),
P24 = Im(S2 S3∗ + S1 S4∗ ),
P31 = Re(S2 S4∗ + S1 S3∗ ),
P32 = Re(S2 S4∗ − S1 S3∗ ),
P33 = Re(S1 S2∗ + S3 S4∗ ),
P34 = Im(S2 S1∗ + S4 S3∗ ),
P41 = Im(S2 S4∗ + S1 S3∗ ),
P42 = Im(S4 S2∗ − S1 S3∗ ),
P43 = Im(S1 S2∗ − S3 S4∗ ),
P44 = Re(S1 S2∗ − S3 S4∗ ).
The P matrix elements contain the full information about the scattering event. In
non absorptive media the elements of the Mueller matrix [Eq. (2.64)] can be used to
define the scattering cross-section and anisotropy. The scattering cross-section σs is
defined as the geometrical cross-section of a scattering object that would produce an
amount of light scattering equal to the total observed scattered power in all directions.
It can be calculated by the integration of the scattered intensity over all directions. It
can be expressed by the elements of the scattering matrix P and the Stokes parameters
(I0 , Q0 ,U0 ,V0 ) of the incident light as follows [40]
σs =
1
2
kh I0
[I0 P11 + Q0 P12 +U0 P13 +V0 P14 ] dΩ.
(2.65)
4π
In the case of a spherically symmetrical scattering object and non polarized light
the relationship (2.65) is reduced to the usual integral of the indicatrix with respect
to the scattering angle:
σs =
2π
kh2
π
P11 (θ ) sin(θ )d θ .
0
The anisotropy parameter g is defined as follows [40]
(2.66)
FDTD Simulation of Light Interaction with Cells
g =< cos(θ ) >=
2π
kh2 σs
π
cos(θ )P11 (θ ) sin(θ )d θ .
19
(2.67)
0
A positive (negative) value of g corresponds to a forward (backward) dominated
scattering. The isotropic scattering case corresponds to g = 0.
In an absorptive medium, the elements of the Mueller scattering matrix [Eq. (2.64)]
depend on the radial distance from the scattering object and can not be directly related to the scattering cross-section as given above by Eq. (2.65). In this case the
different elements of the matrix are used individually in the analysis of the scattering
phenomena. In practice, their values are normalized by the total scattered rate around
the object in the radiation zone, which can be derived from the integral of P11 for all
scattering angles.
Absorption, scattering and extinction efficiencies
The flow of energy and the direction of the electromagnetic wave propagation are
represented by the Poynting vector:
∗,
s = E × H
(2.68)
where the asterisk denotes the complex conjugate. To derive the absorption and
extinction rates of a particle embedded in an absorptive medium, we can rewrite the
last equation in Eq. (2.46) as
= −iωε0 (εr + iεi )E.
∇×H
(2.69)
Combining the third equation in Eqs. (2.46) with Eq. (2.69) leads to
∗) = H
∗ · (∇×E) − E · (∇×H
∗)
∇ · s =∇ · (E × H
·H
∗ −ε 0 εr E · E ∗ ) − ωε 0 εi E · E ∗
= iω (µ0 H
(2.70)
For the sake of convenience in the following presentation, we will define the real
and imaginary parts of the relative permittivity of the scattering object as εtr and
εti , and those for the host medium as εhr and εhi , respectively. The rate of energy
absorbed by the object is
!
"
wa = − 12 Re S n ·s(ξ )d 2ξ = − 12 Re V ∇ ·s(ξ )d 3ξ
(2.71)
"
= ε0 ω V εti (ξ )E(ξ ) · E ∗ (ξ )d 3ξ ,
2
where n denotes the outward-pointing unit vector normal to the surface of the object.
The surface and volume integrals are defined by the volume of the scattering object.
When electromagnetic waves are incident on an object, the electric and magnetic
can be taken as sums of the incident and scattered fields.
field vectors E and H
Therefore the scattered field vectors can be written as
Es = E − E i ,
(2.72)
20
Handbook of Photonics for Biomedical Science
−H
i,
s= H
H
(2.73)
i denote the incident electric and magnetic field vector, respectively.
where Ei and H
Therefore the rate of energy scattered by the object can be expressed as
$#
%
s∗ )d 2ξ = 1 Re
∗ −H
i∗ ) d 2ξ .
n · (Es × H
n · (E − Ei ) × (H
2
S
S
(2.74)
Because both absorption and scattering remove energy from the incident waves,
the extinction rate of the energy can be defined as
ws =
1
Re
2
#
we = ws +wa =
!
!
∗ −H
i∗ )]d 2ξ }− 1 Re[ S n · (E×H
∗ )d 2ξ ]
= 12 Re{ S n · [(E − E i )×(H
2
!
∗
∗
∗
i −E i ×H
−E × H
i )d 2ξ ]
= 12 Re[ S n · (E i ×H
"
1
∗
∗
= 2 Re[ V Ñ × (E i ×H i −E i ×H −E × H ∗i )d 3 ξ ].
(2.75)
Using Eqs. (2.70) and (2.75), similar to the derivation of Eqs. (2.71), we can
obtain
"
3
we = wa +ws = ε 0 ω2 V [ε ti (ξ ) + ε hi (ξ )] Re[E i (ξ ) · E ∗ (ξ )]d ξ
"
3
−ε 0 ω2 V [ε tr (ξ ) − ε hr (ξ )] Im[E i (ξ ) · E ∗ (ξ )]d ξ
"
−ε 0 ω εhi (ξ )[E (ξ ) · E ∗ (ξ )]d 3ξ .
2 V
(2.76)
i
i
Assuming the rate of energy incident on a particle of arbitrary shape is f , then the
absorption, scattering and extinction efficiencies are Qa = wa / f , Qs = (we −wa )/ f
and Qe = we / f , respectively. Consequently, the single scattering albedo is ω̃ = Qs /Qe .
In an absorptive medium, the rate of energy incident on the object depends on the
position and intensity of the wave source, the optical properties of the host medium,
and the objects size and shape. For spherical objects, if the intensity of the incident
light at the center of the computational domain is I0 , the rate of energy incident on a
spherical scatterer centered at the center of the 3D computational domain is
f=
2π a2
I0 [1 + (η − 1)eη ],
η2
(2.77)
where a is the radius of the spherical object, η = 4π anhi /λ0 , I0 = 12 cnµhr0 |E0 |2 is
the intensity of the incident light at the center of the computational domain, λ0 is the
incident wavelength in free space, nhr and nhi are the real and imaginary refractive
index of the host medium, respectively. |E0 | is the amplitude of the incident electric
field at the center of the 3D computational domain. For non-spherical object, the rate
of energy incident on the object is calculated numerically.
FDTD Simulation of Light Interaction with Cells
21
FIGURE 2.4: Schematic representation of the modified 3D FDTD Total
Field/Scattered Field formulation.
2.2.5
FDTD formulation of optical phase contrast microscopic (OPCM)
imaging
The 3D FDTD formulation provided here is based on a modified version of the
total-field/scattered-field (TFSF) formulation that was described earlier [29, 32]. It
could be more appropriately called total-field/reflected-field (TFRF) formulation.
The 3D TFRF formulation uses a TFSF region which contains the biological cell
and extends beyond the limits of the simulation domain (Fig. 2.4). The extension
of the transverse dimension of the input field beyond the limits of the computational
domain through the UPML boundaries would lead to distortions of its ideal plane
wave shape and eventually distort the simulation results. To avoid these distortions
one must use Bloch periodic boundary conditions (Fig. 2.4) in the lateral x- and
y-directions which are perpendicular to the direction of propagation z [29].
Bloch boundary conditions are periodic boundary conditions which take into account the phase effects due to the tilting of the input plane waves incoming at periodic
structures, i.e. what we are actually modeling is a periodic row of biological cells.
The near scattered fields, however, are calculated in the transverse planes located in
the close proximity to the cell where the coupling effect due to waves scattered from
adjacent cells is negligible. This effect can be further minimized or completely removed by controlling the lateral dimension of computational domain by using a large
enough period of the periodic cell structure. The larger is this period, the smaller is
22
Handbook of Photonics for Biomedical Science
the coupling effect. In the 3D TFRF formulation the location in the computational
domain corresponding to the forward scattered light is positioned within the total
field region (Fig. 2.4). The OPCM simulation model requires the explicit availability of the forward scattered transverse distribution of the fields. The phase of the
scattered field accumulated by a plane wave propagating through a biological cell
will be used in the FDTD model of the OPCM that will be described in the next
section.
FDTD OPCM principle
Phase contrast microscopy is utilized to produce high-contrast images of transparent specimens such as microorganisms, thin tissue slices, living cells and sub-cellular
components such as nuclei and organelles. It translates small phase variations into
corresponding changes in amplitude visualized as differences in image contrast. A
standard phase contrast microscope design is shown in Fig. 2.5a, where an image
with a strong contrast ratio is created by coherently interfering a reference (R) with
a diffracted beam (D) from the specimen.
Relative to the reference beam, the diffracted beam has lower amplitude and is
retarded in phase by approximately π /2 through interaction with the specimen. The
main feature in the design of the phase contrast microscope is the spatial separation
of the R beam from D wave front emerging from the specimen. In addition, the
amplitude of the R beam light must be reduced and the phase advanced or retarded
by another ±π /2 in order to maximize the differences in the intensity between the
specimen and the background in the image plane. The mechanism for generating
relative phase retardation has two-steps: i) the D beam is being retarded in phase by
a quarter wavelength (i.e., π /2) at the specimen, and ii) the R beam is advanced (or
retarded) in phase by a phase plate positioned in or very near the objective rear focal
plane (Fig. 2.5a). This two-step process is enabled by a specially designed annular
diaphragm – the annulus. The condenser annulus, which is placed in the condenser
front focal plane, is matched in diameter and optically conjugated to the phase plate
residing in the objective rear focal plane. The resulting image, where the total phase
difference is translated by interference at the image plane into an amplitude variation,
can have a high contrast ratio, particularly if both beams have the same amplitude.
Fig. 2.5a illustrates the part of the microscope that will become the subject of
FDTD modeling combined with Fourier optics. Fig. 2.5b provides a visual representation illustrating the major steps in the FDTD OPCM model. The phase contrast
microscope uses incoherent annular illumination that could be approximately modeled by adding up the results of eight different simulation using ideal input plane
waves incident at a given polar angle (30 deg), an azimuthal angle (0, 90, 180 or
270 deg), and a specific light polarization (parallel or perpendicular to the plane of
the graph). Every single FDTD simulation provides the near field components in a
transverse monitoring plane located right behind the cell (Fig. 2.4).
The far field transformations use the calculated near fields right behind the cell
and return the three complex components of the electromagnetic fields far enough
from the location of the near fields, i.e. in the far field [29]: Er (ux , uy ), Eθ (ux , uy )
FDTD Simulation of Light Interaction with Cells
23
FIGURE 2.5: a) Schematic representation of an OPCM. b) 2D visual representation of the FDTD OPCM model using incoherent illumination by two planes waves
at a polar angle of 30 deg. For each of the two plane waves the propagation of light
is modeled as a combination of two parallel wave phenomena: i) propagation of the
reference (R) beam without the cell, and ii) propagation of the diffracted (D) beam
due to the cell.
and Eφ (ux , uy ), where r, θ and φ refer to the spherical coordinate system shown in
Fig. 2.3, and the variables ux and uy are the x and y components of the unit direction
vector u. The unit direction vector u is related to the angular variables θ and φ :
ux = sin(θ ) cos(φ ),
uy = sin(θ ) sin(φ ),
uz = cos(θ ),
u2x +u2y +u2z = 1. (2.78)
The in-plane wave vectors for each plane wave are given by kx = kux and ky = kuy ,
where k = 2π /λ . What is important here is to note that in this case the near-to-far
field transformation must take into account the Bloch periodic boundary conditions
in the lateral dimension and is calculated only at angles that correspond to diffracted
orders of the periodic structure such as defined by the Bragg conditions. This is
24
Handbook of Photonics for Biomedical Science
done by calculating the direction cosines of all the diffracted orders that meet the
Bragg condition, and interpolating the previously far-field distributions onto those
specific directions. The near-to-far field projection therefore provides the electric
field amplitude and phase corresponding to each diffracted order. The zeroth order,
i.e. the light travelling through the scattering object without any deviation in angle, is
the reference beam and the phase contrast microscope is designed to provide a phase
delay to this order.
The amplitudes and the phases of the calculated far-field components can now
be used to do Fourier optics with both the scattered and reference beams. We can
assume an ideal optical lens system that could be characterized by a given magnification factor. This simple model could be easily extended to include the numerical
equivalent of the two lenses together with an additional model to take into account
any aberrational effects. The magnification was implemented by modifying the angle of light propagation, i.e. by multiplying the transverse components of the direction cosines, ux and uy by the inverse value of the desired magnification factor M:
Ux = ux /M and Uy = uy /M. In any other circumstances the modification of the direction cosines would lead to complications because of the vectorial nature of the E
field. In our case, however, working in spherical coordinates (Er , Eθ and Eφ ) leads
to the advantage that the vectorial components do not change when ux and uy are
modified because they are part of a local coordinate system that is tied to the values
ux and uy . The factor M is applied to the far fields before the interference of the
diffracted (D) and reference (R) beams at the image plane.
It was also possible to apply the effect of a numerical aperture NA which clips any
light that has too steep an angle and would not be collected by the lens system. This
means that all the light with Ux2 + Uy2 > (NA)2 is being clipped. The effect of the
aperture is defined by applying the last inequality to the corrected aperture angles θ and φ :
sin(φ ) = Uy /Uxy ,
cos(φ ) = Ux /Uxy ,
cos(θ ) = Uz ,
sin(θ ) = Uxy ,
(2.79)
2 ) and the “sqrt” labels a square root
where Uxy = sqrt(Ux2 + Uy2 ), Uz = sqrt(1 − Uxy
mathematical operation. The magnified field components will then have the following form:
Diffracted (D) beam:
Ex−D (kx , ky ) = −Eφ −D sin(φ ) + Eθ −D cos(φ ) cos(θ ),
Ey−D (kx , ky ) = Eφ −D cos(φ ) + Eθ −D sin(φ ) cos(θ ),
Ez−D (kx , ky ) = −Eθ −D sin(θ ).
Reference (R) beam:
Ex−R (kx , ky ) = −Eφ −R sin(φ ) + Eθ −R cos(φ ) cos(θ ),
(2.80)
FDTD Simulation of Light Interaction with Cells
Ey−R (kx , ky ) = Eφ −R cos(φ ) + Eθ −R sin(φ ) cos(θ ),
25
(2.81)
Ez−R (kx , ky ) = −Eθ −R sin(θ ),
where Eθ −R and Eφ −R are the far field components of the reference beam.
The fields given above are then used to calculate back the Fourier inverse transform of the far field transformed fields leading to the distribution of the scattered and
the references beams in the image plane:
Diffracted (D) beam:
Ex−D = sum (Ex−D (kx , ky ) exp(ikx x + iky y + ikz z)) ,
Ey−D = sum (Ey−D (kx , ky ) exp(ikx x + iky y + ikz z)) ,
(2.82)
Ez−D = sum (Ez−D (kx , ky ) exp(ikx x + iky y + ikz z)) .
Reference (R) beam:
Ex−R = sum (Ex−R (kx , ky ) exp(ikx x + iky y + ikz z)) ,
Ey−R = sum (Ey−R (kx , ky ) exp(ikx x + iky y + ikz z)) ,
(2.83)
Ez−R = sum (Ez−R (kx , ky ) exp(ikx x + iky y + ikz z)) ,
where the summation is over all angles.
The OPCM images at the image plane are calculated by adding up the scattered
and the reference beam at any desired phase offset Ψ:
I = abs(Ex−D + aEx−R exp(iΨ))2 + abs(Ey−D + aEy−R exp(iΨ))2
+ abs(Ez−D + aEz−R exp(iΨ))2 .
(2.84)
The coefficient a and the phase Ψ are simulation parameters corresponding to
the ability of the OPCM to adjust the relative amplitudes and the phase difference
between the two beams.
2.3 FDTD Simulation Results of Light Scattering Patterns From
Single Cells
2.3.1
Validation of the simulation results
To simulate the light scattering and absorption by single biological cells, we use a
C++ computer program that is based on the FDTD formulation described in section
26
Handbook of Photonics for Biomedical Science
2.2.4 [17, 32]. A Mie scattering program was also used to provide exact results for
the validation of the FDTD simulations. Details about the specific Mie scattering
formulation can be found in [27] and [42].
Figure 2.6 shows some preliminary FDTD simulation results for the phase function of a simple spherical biological cell containing only a cytoplasm and a nucleus
both embedded in a non-absorptive extra-cellular medium. The phase function represents the light scattering intensity as a function of the scattering angle in a plane
passing through the center of the cell. It is defined by the angular dependence of the
P11 element of the Mueller scattering matrix normalized by the total scattered rate
- the integral of P11 in all possible directions around the cell. The cell membrane
is not taken into account. The nucleus has a size parameter 2π Rc /λ0 = 7.2498,
where Rc is the radius of the nucleus and and λ0 is the incident wavelength in free
space. The refractive index of the nucleus is 1.4, of the cytoplasm 1.37 and of
the extra-cellular material 1.35. The size parameter of the whole cell is 19.3328.
The FDTD cell size is ∆s = λ0 /30 and the UPML parameters are κmax = 1 and
R(0) = 10−8 [see Eqs. (2.12) and (2.13)]. The number of mesh points in all three
directions is the same: 209. The number of simulation time steps is 10700. These
preliminary results were compared with the exact solutions provided by Mie theory.
The relative error of the FDTD simulation results is ∼5%. Some absolute values
of cell parameters can be derived as follows. If λ0 = 0.9µ m, the FDTD cell size
∆s = λ0 /30 = 0.03µ m, the nucleus’ radius Rc = 7.2498λ0 /2π = 1.0385µ m and the
cytoplasm radius Rc = 19.3328λ0 /2π = 2.7692µ m. These dimensions are more typical for relatively small cells or bacteria which have no nucleus.
The extinction efficiency Qe , absorption efficiency Qa and anisotropy factor g for
the cell associated with Fig. 2.6 with the parameters given above are listed in Table
2.1. Other P matrix elements are shown in Fig. 2.7. To complete the validation of the
UPML FDTD scheme we compared the numerical and exact solutions for the light
scattering patterns from biological cells embedded in an absorptive extra-cellular
medium.
TABLE 2.1:
Comparison between FDTD simulation results and analytical
solutions provided by Mie theory for the extinction efficiency Qe , scattering
efficiency Qs , absorption efficiency Qa and anisotropy factor g of a cell with a
cytoplasm and a nucleus in non-absorptive extra-cellular medium
Results
Mie theory
FDTD
(FDTD-Mie)/Mie
Qe
0.359057
0.358963
0.026 %
Qs
0.359057
0.358963
0.026 %
Qa g
0.00.993889
0.00.993914
- 0.0025 %
Accounting for this absorption effect requires a special attention since not all types
FDTD Simulation of Light Interaction with Cells
27
FIGURE 2.6: Normalized light scattering intensity (P11 element of the Mueller
scattering matrix) distribution with scattering angle – comparison of the FDTD simulation results with exact analytical (Mie theory) results. There is a very good agreement between exact analytical and the numerical solutions with a relative error of
approximately 5%.
of boundary conditions can handle absorptive materials touching the boundary of
the simulation domains. One of the advantages of the UPML boundary conditions
considered here is that they can handle that [17]. To study the effect of the absorption
of the extra-cellular medium we assume that the refractive index of the extra-cellular
medium has a real and an imaginary part: n + ik. Figure 2.8 demonstrates the good
agreement (∼5% relative error) between the FDTD and the exact (Mie theory) results
for the normalized light scattering patterns in the case when the refractive index the
extra-cellular medium is 1.35 + i0.05.
The extinction efficiency Qe , scattering efficiency Qs , absorption efficiency Qa
and anisotropy factor (g) for a cell in an absorptive medium (the same as in Fig. 2.8)
are listed in Table 2.2.
2.3.2
Effect of extra-cellular medium absorption on the light scattering
patterns
This section describes the FDTD simulation results for the effect of absorption in
the extra-cellular medium on the light scattering patterns from a single cell [17]. We
consider two different cell geometries: the one considered in the previous section
and another one with a shape of a spheroid (both cytoplasm and nucleus) which can
be described by the surface function
x2 y2 z2
+ + = 1,
a2 b2 c2
(2.85)
28
Handbook of Photonics for Biomedical Science
FIGURE 2.7: Angular distributions of the normalized scattering matrix elements
P12 , P33 , P43 and P44 calculated by the FDTD method. Cell parameters are the same
as for Fig. 2.6. There is a very good agreement between exact (Mie theory) and
numerical results with a relative error of the FDTD results of approximately 5%.
TABLE 2.2:
Comparison between FDTD simulation results and the
analytical solutions provided by Mie theory for the extinction efficiency Qe ,
scattering efficiency Qs , absorption efficiency Qa and anisotropy factor g of a
cell with a cytoplasm and a nucleus in a non-zero imaginary part of the
refractive index of the extra-cellular medium k = 0.05 and cell parameters as
described above in this section.
Results
Mie theory
FDTD
(FDTD-Mie)/Mie
Qe
0.672909
0.676872
0.589 %
Qs
0.672909
0.676872
0.589 %
Qa
0.0
0.0
-
g
0.992408
0.992440
0.0032 %
where a, b, and c are the half axes of the spheroid with a = 2b = 2c.
Figure 2.9 illustrates the light scattering configuration in the case of a cell with
a spheroid shape. The light is incident in the x-direction along the long axis of the
spheroid. The size parameters of the cytoplasm are defined by 2π Ra /λ0 = 40 and
2π Rb,c /λ0 = 20, where Ra = a/2 and Rb,c = b/2 = c/2. The size parameters of the
FDTD Simulation of Light Interaction with Cells
29
FIGURE 2.8: Normalized light scattering intensity distribution with scattering angle in the case of absorptive extra-cellular medium. Cell parameters are the same as
for Fig. 2.6 and Fig. 2.7, except a non-zero imaginary part of the refractive index of
the extra-cellular medium.
FIGURE 2.9: Coordinate system and geometry of a cell with the shape of a
spheroid with half axes a, b, and c, where a = 2b = 2c. The light is incident in
the positive x-direction, along the long axis a.
nucleus are defined by 2π ra /λ0 = 20 and 2π rb,c /λ0 = 10, where ra and rb are the
large and small radii of the nucleus, respectively. The cell refractive indices are:
cytoplasm: 1.37; nucleus: 1.40, extra-cellular medium: 1.35, and 1.35+0.05i. The
FDTD cell size is ∆z = λ /20. The number of mesh point are Nx = 147, Ny = 147 and
Nz = 275. The number of simulation time steps is 14000. Figure 2.10 shows: i) the
effect of cell size on the light scattering patterns, and ii) the effect of the absorption
of the extra-cellular medium on the phase function of spheroid cells [16–17].
A more detailed analysis of the two graphs shown in Fig. 2.10 leads to some interesting findings. First, absorption in the extra-cellular material of spherical cells (Fig.
2.10, right graph) increases the intensity of the light scattering up to one order in
the angle range between 90o (transverse scattering) and Ψ = 180o (backward scattering). The same light scattering feature was also found in the case of spheroid cells.
30
Handbook of Photonics for Biomedical Science
FIGURE 2.10: Normalized light scattering intensity distribution with scattering
angle for two different cases: absorptive and non-absorptive extra-cellular medium
for spherical (left) and spheroid (right) cells. The values of the imaginary part of the
refractive index of the extra-cellular medium are indicated in the insight.
Second, the influence of absorption in the extra-cellular material is relatively more
pronounced in the case of spherical as compared to spheroid cells (Fig. 2.10, left
graph). Third, in the case of exact backward scattering and absorptive extra-cellular
material, the light scattering intensity is approximately equal for both cell shapes.
However this is not true for non-absorptive cell surroundings. This last finding could
be highly relevant for OCT and onfocal imaging systems, especially for studies in
the wavelength ranges within the hemoglobin, water and lipids bands.
These findings show that the analysis of light scattering from isolated biological
cells should necessarily account for the absorption effect of the surrounding medium.
It could be particularly relevant in the case of optical immersion techniques using
intra-tissue administration of appropriate chemical agents with absorptive optical
properties [43-45]; however, this relevance has not been studied before. It should
be pointed out that whenever there is a matching of the refractive indices of a light
scatterer and the background material, the scattering coefficient goes to zero and it is
only the absorption in the scatterer or in the background material that will be responsible for the light beam extinction. The results presented here provide some good
preliminary insights about the light scattering role of absorption in the background
material; however, it needs to be further studied.
2.4 FDTD Simulation Results of OPCM Nanobioimaging
2.4.1
Cell structure
This section describes the 3D FDTD modeling results of OPCM imaging of single
biological cells in a number of different scenarios. The results are based on the FDTD
FDTD Simulation of Light Interaction with Cells
31
OPCM model described in subsection 2.2.5 [46]. The optical magnification factor
M = 10 and the numerical aperture NA = 0.8. The cell is modeled as a dielectric
sphere with a realistic radius Rc = 5µ m (Fig. 2.4). The cell membrane thickness
is d = 20nm which corresponds to effective (numerical) thickness of approximately
10 nm. The cell nucleus is also spherical with a radius Rn = 1.5µ m centered at a
position which is 2.0 µ m shifted from the cell center in a direction perpendicular to
the direction of light propagation. The refractive index of the cytoplasm is ncyto =
1.36, of the nucleus nnuc = 1.4, of the membrane nmem = 1.47 and of the extracellular material next = 1.33 (no Refractive Index Matching – no RIM) or 1.36 (RIM).
2.4.2
Optical clearing effect
The RIM between the cytoplasm and the extra-cellular medium leads to the optical
clearing of the cell image. The optical clearing effect leads to the increased light
transmission through microbiological objects due to the matching of the refractive
indices of some of their components to that of the extra-cellular medium [42–45]. If
a biological object is homogenous, matching its refractive index value by externally
controlling the refractive index of the host medium will make it optically invisible.
If the biological object contains a localized inhomogeneity with a refractive index
different from the rest of the object, matching the refractive index of the object with
that of the external material will make the image of the object disappear and sharply
enhance the optical contrast of the inhomogeneity. In the case of biological cells
the refractive index of the extra-cellular fluid can be externally controlled by the
administration of an appropriate chemical agent [42–45].
Figure 2.11 shows the cross-sections of two cell images for different values of
the phase offset Ψ between the reference and diffracted beam of the OPCM: 180◦
and 90◦ . The images illustrate the nature of the optical clearing effect and the value
of its potential application for the early detection of cancerous cells by a careful
examination of their nucleus size, eccentricity, morphology and chromatin texture
(refractive index fluctuations) [14]. At no RIM conditions in both cases (Ψ = 180◦
and Ψ = 90◦ ) the image of the nucleus is represented by a dip in the cell image. At
RIM conditions the image contrast of the cell is drastically reduced to zero levels
and it is only the image of the nucleus that remains sharply visible. The image of
the nucleus is represented by a nice peak associated with the 3-dimensional optical
phase accumulation corresponding to its perfectly spherical shape and homogeneous
refractive index distribution. A finer analysis of the two graphs shown in Fig. 2.11
will show that the diameter of the nucleus (the full width at the half-height of the nucleus image contrast peak) depends on the phase delay Ψ. At Ψ = 180◦ and no RIM
conditions the diameter of the nucleus is estimated at value of ∼2.3 µ m as compared
to RIM conditions where it’s value is 3.3 µ m (the estimation accounts for the optical
magnification factor ×10 of the system). At Ψ = 90◦ and no RIM conditions the
diameter of the nucleus is estimated at a value of ∼3.05 µ m as compared to RIM
conditions where its value is 3.75 µ m (the cell model used in the FDTD simulations
has a nucleus with a diameter 3.0 µ m. This shows that the OPCM should be preliminary set-up at a given optimum phase delay and the OPCM images should be used
32
Handbook of Photonics for Biomedical Science
FIGURE 2.11: Cross-sections of FDTD-generated OPCM images of a single cell
illustrating the optical clearing effect for different values of the phase offset Ψ between the reference and diffracted beam of the OPCM: 180◦ (on the left-hand side)
and 90◦ (on the right-hand side). Matching the refractive index value of the extracellular material with that of the cytoplasm enhances the optical contrast and leads
to a finer view of the morphological structure of the nucleus.
for relative measurements only after a proper calibration. The analysis of the graphs,
however, shows an unprecedented opportunity for using the optical clearing effect
for the analysis of any pathological changes in the eccentricity and the chromatin
texture of cell nuclei within the context of OPCM cytometry configurations. This
new opportunity is associated with the fact that at RIM the cell image is practically
transformed into a much finer image of the nucleus.
2.4.3
The cell imaging effect of gold nanoparticles
Optical properties of gold nanoparticles (NPs)
Gold NPs have the ability to resonantly scatter visible and near infrared light. The
scattering ability is due to the excitation of Surface Plasmon Resonances (SPR). It is
extremely sensitive to their size, shape, and aggregation state offering a great potential for optical cellular imaging and detection labeling studies [47–51]. Our FDTD
approach [21, 24] uses the dispersion model for gold derived from the experimental
data provided by Johnson and Christy [52] where the total, complex-valued permittivity is given as:
ε (ω ) = εreal + εL (ω ) + εP (ω ).
(2.86)
Each of the three contributions to the permittivity arises from a different material model. The first term represents the contribution due to the basic, background
permittivity. The second and third terms represent Lorentz and plasma contributions:
εL (ω ) = εLorentz ω02 /(ω02 − 2iδ0 ω − ω 2 ),
εP (ω ) = ωP2 /(iωνC + ω 2 ),
(2.87)
FDTD Simulation of Light Interaction with Cells
33
where all material constants are summarized in Table 2.3.
TABLE 2.3:
Background
permittivity
εreal = 7.077
Optical material constants of gold [52]
Lorentz dispersion
Plasma dispersion
εLorentz = 2.323
ω0 = 4.635 × 1015 Hz
δ0 = 9.267 × 1014 Hz
ωP = 1.391 × 1016 Hz
νC = 1.411 × 107 Hz
We have modeled both the resonant and non-resonant cases. The ability to model
these two different cases, together with the effect of optical clearing effect, provides
the opportunity to numerically study the possibility for imaging the uptake of clusters
of NPs – a scenario which needs to be further studied [21]. We have also used
the FDTD technique to calculate the scattering and absorption cross-sections over
a 400–900 nm wavelength range for a single 50 nm diameter gold NP immersed
in a material having the properties of the cytoplasm (ncyto = 1.36) and resolution
dx = dy = dz = 10 nm. The scattering cross-section is defined as
σscat = Pscat (ω )/Iinc (ω ),
(2.88)
where Pscat is the total scattered power and Iinc is the intensity of the incident light
in W / m2 . It was calculated by applying the total-field/scattered-field FDTD formulation described in subsection 2.2.2. We have used the GUI features of the FDTD
Solutions software to create 12 field power monitoring planes around the nanoparticle in the form of a box: 6 in the total field region and 6 in the scattered field region.
The total scattered power was calculated by summing up the power flowing outward
through 6 scattered field power monitors located in the scattered field region.
The absorption cross-section is similarly defined as
σabs = Pabs (ω )/Iinc (ω ),
(2.89)
where Pabs is the total power absorbed by the particle. The power absorbed by the
particle is calculated by calculating the net power flowing inward through the 6 total
field power monitors located in the total field region.
The extinction cross-section is the sum of the absorption and scattering crosssections
σext = σscat + σabs .
(2.90)
Figure 2.12a shows that the extinction cross-section has a maximum of 3.89 at
around 543.0 nm corresponding to one of the radiation wavelengths of He-Ne lasers.
Here we also present the results for λ = 676.4 nm (a Krypton laser wavelength)
34
Handbook of Photonics for Biomedical Science
FIGURE 2.12: a) Extinction cross-section of a 50 nm gold nanoparticle immersed
in material having the optical properties of the cytoplasm n = 1.36 (left). The gold
optical properties are described by Eqs. (2.86), (2.87) with the parameters given
in Table 2.3. b) Positioning of a cluster of gold nanoparticles randomly distributed
within a spatial sphere at a cell location symmetrically opposite to the center of the
nucleus.
which corresponds to the non-resonant case (extinction cross-section value 0.322,
∼12 times smaller than 3.89). The FDTD results are compared with the theoretical
curve calculated by Mie theory. The slight discrepancy between the theoretical and
FDTD results for the extinction cross-section is due to the finite mesh size. The
consistency of the results could be visibly improved by reducing the mesh size.
OPCM images of gold nanoparticle clusters in single cells
The OPCM cell images are the result of simulations using non-uniform meshing
where the number of mesh points in space is automatically calculated to ensure a
higher number of mesh points in materials with higher values of the refractive index
[53]. Fig. 2.12b visualizes the schematic positioning of a cluster of 42 nanoparticles in the cytoplasm that was used to produce the simulation results presented in
this section. The cell center is located in the middle (x = y = z = 0) of the computational domain with dimensions 15 µ m × 12 µ m × 15 µ m (Fig. 2.12b). The
center of the nucleus is located at x = −2 µ m, y = z = 0 µ m. The cluster of gold
nanoparticles is located at x = 2 µ m, y = z = 0 µ m. The realistic cell dimensions
(including both cell radius and membrane) require a very fine numerical resolution
making the simulations computationally intensive. The numerical resolution of the
nanoparticles was hard-coded to dx = dy = dz = 10 nm to make sure that their numerically manifested optical resonant properties will be the same as the ones shown
in Fig. 2.12a. This lead to additional requirements for the CPU time and memory (∼120 Gbs RAM) requiring high performance computing resources. The time
FDTD Simulation of Light Interaction with Cells
RIM, no NPs,
676.4nm
RIM, NPs,
676.4nm
35
RIM, NPs,
543.0nm
FIGURE 2.13: (Color figure follows p. .) OPCM images of a single cell for
different values (a: −150o , b: −90o , c: +90o , d: +180o ) of the phase offset Ψ
between the reference and diffracted beam of the OPCM at RIM (optical immersion)
conditions including a cluster of 42 gold NPs located in a position symmetrically
opposite to the nucleus. The arrows indicate the position of the cluster. The left
column corresponds to a cell without NPs. The other columns correspond to a cell
with NPs at resonant (right) and non-resonant (middle) conditions.
step used during the simulation was defined by means of the Courant stability limit:
−1/2
.
c∆t = 0.99 × 1/∆x2 + 1/∆y2 + 1/∆z2
Based on the fact that RIM enhances significantly the imaging of the cell components, we have used the FDTD OPCM model to create the OPCM images of the cell
36
Handbook of Photonics for Biomedical Science
FIGURE 2.14: Comparison of the geometrical cross sections at y = 0 µ m of the
three OPCM images corresponding to a phase offset Ψ = 180o between the reference
and the diffracted beam (bottom row in Fig. 2.13) in terms of optical contrast. The
right hand-side graph illustrates the optical contrast enhancement due to the effect of
the gold NP resonance at λ = 543.0 nm.
FIGURE 2.15: Optical contrast due to the gold NP cluster as a function of the
phase offsets between the reference (R) and the diffracted (D) beams of the optical
phase microscope.
FIGURE 2.16: A cluster of 42 Gold NPs randomly distributed on the surface of
the cell nucleus. The NP size on right hand graph is slightly exaggerated.
FDTD Simulation of Light Interaction with Cells
RIM, no NPs,
676.4nm
RIM, NPs,
676.4nm
37
RIM, NPs,
543.0nm
FIGURE 2.17: OPCM images of the cell for different values of the phase offset Ψ
(a: −90o , b: −30o , c: +30o , d: +90o ) between the reference and diffracted beam of
the OPCM at optical immersion, i.e. refractive index matching, conditions without
NPs (left) and including 42 Gold NPs (middle – at no resonance, right – at resonance)
randomly located at the surface of the cell nucleus.
at optical immersion conditions including the cluster of 42 gold NPs (Fig. 2.13) and
for different values of the phase offset Ψ between the reference beam and the scattered beam [assuming a = 1, see Eq. (2.62)]. The two graphs in Fig. 2.14 compare
the geometrical cross sections (y = 0µ m) of the three OPCM images shown at the
bottom of Fig. 2.13. The right-hand side graph shows the relevant half of the image
where the gold NP cluster is located.
At resonance the optical contrast of the gold NP peak is ∼2.24 times larger than
the one at no resonance and 24.79 times larger than the background optical contrast
38
Handbook of Photonics for Biomedical Science
FIGURE 2.18: Cross-sections of the images shown in Fig. 2.17c (corresponding
to phase offset Ψ = +30o ) expressed in terms of optical contrast. The right-hand
side graph provides in finer details an enlarged portion of the left-hand side one. The
specific fragmentation of the nucleus’ image is due to the presence of the Gold NPs
at resonant condition.
corresponding to the case when there are no nanoparticles. The enhanced imaging of
the gold NP cluster at resonant conditions is clearly demonstrated. It however needs
to be further studied as a function of particular phase offset Ψ between the reference
beam and the scattered beam.
Figure 2.15 visualizes the optical contrast due to the gold nanoparticle cluster as a
function of the phase offsets between the reference (R) and the diffracted (D) beams
of the optical phase microscope [21]. It shows that the enhancement of the optical
contrast due to the nanoparticle resonance changes significantly from minimum of
0.0 (Ψ = 0o ) to a maximum of 3.60 (Ψ = −150o ). This finding should be taken into
account in real life OPCM imaging experiments.
OPCM images of gold nanoparticles randomly distributed on the nucleus of a
cell
Figure 2.16 visualizes the positioning of a cluster of 42 Gold NPs on the surface
of the cell nucleus. This was the NP configuration used to produce the FDTD-based
simulation results presented in this section [24].
It should be pointed out that the optical wave phenomena involved in the simulation scenario considered here are fundamentally different from the ones considered
in the previous section where the gold nanoparticles are randomly distributed within
the homogeneous material of cytoplasm and their presence is manifested by means
of their own absorption and scattering properties. In the present case the NPs are
located at the interface of the nucleus and the cytoplasm which is characterized by
a relatively large refractive index difference (∆n = 0.04) and which is, therefore,
expected to largely dominate and modify the visual effect of the NPs.
A close examination of the OPCM images in Fig. 2.17 provides an illustration
of this fact. The OPCM cell images are for different values of the phase offset Ψ
FDTD Simulation of Light Interaction with Cells
39
(a: −90o , b: −30o , c: +30o , d: +90o ) at optical immersion and both resonant and
non-resonant conditions. The analysis of Figs. 2.17 and 2.18 leads to a number of
interesting findings.
First, the images of the cell without the Gold NPs are hardly distinguishable from
the images including the Gold NPs at no resonance conditions (λ = 676.4nm). Second, the visual effect of the Gold NP presence at resonant conditions (λ = 543.0 nm)
depends significantly on the phase offset Ψ. Third, the presence of the Gold NPs at
resonant conditions can be identified by a specific fragmentation of the image of the
nucleus for specific values of the offset Ψ (see the right-hand side graphs in Figs.
2.17c and 2.18). This last finding indicates the importance of the ability to adjust the
offset Ψ between the reference and diffracted beam of the OPCM in practical circumstances. It is expected to be of relevance for the development and calibration of
similar optical diagnostics techniques by medical photonics researchers and clinical
pathologists.
2.5 Conclusion
In this chapter we provided a detailed summary of the mathematical formulation of
the FDTD method for application in medical biophotonics problems. We have then
applied the FDTD approach to three different modeling scenarios: i) light scattering
from single cells, ii) OPCM imaging of realistic size cells, and iii) OPCM imaging
of gold NPs in singe cells. We have demonstrated the FDTD ability to model OPCM
microscopic imaging by, first, reproducing the effect of optical immersion on the
OPCM images of a realistic size cell containing a cytoplasm, a nucleus and a membrane. Second, the model was applied to include the presence of a cluster of gold
NPs in the cytoplasm at optical immersion conditions as well as the enhancing imaging effect of the optical resonance of the nanoparticles. Third, we have studied the
imaging effect of gold NPs randomly distributed on the surface of the cell nucleus.
The results do not allow analyzing the scaling of the NP imaging effect as a function
of the number of the NPs. However, the validation of the model provides a basis for
future research on OPCM nanobioimaging including the effects of NP cluster size,
NP size and number, as well as average distance between the NPs. Another future
extension of this research will be to study the capability of the model to provide
valuable insights for the application of gold NPs in optical nanotherapeutics.
We believe that the shift from the modeling of the light scattering properties of
single cells to the construction of OPCM images of cells containing gold NPs represents a major step forward in extending the application of the FDTD approach to
biomedical photonics. It opens a new application area with a significant research
potential – the design and modeling of advanced nanobioimaging instrumentation.
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Handbook of Photonics for Biomedical Science
Acknowledgement
ST and JP acknowledge the use of the computing resources of WestGrid (Western
Canada Research Grid) - a $50-million project to operate grid-enabled high performance computing and collaboration infrastructure at institutions across Canada.
VVT was supported by grants No. 224014 PHOTONICS4LIFE-FP7-ICT-2007-2
(2008-2013), RF President’s “Supporting of Leading Scientific Schools” - NSHA208.2008.2 (2008-2009), and Nos. 2.1.1/4989, 2.2.1.1/2950 of PF Program on the
Development of High School Potential (2009-2010).
We are all thankful to Dr. Vladimir P. Zharov for a preliminary discussion on the
relevance of the problem of nanoparticle clustering in single cells.
FDTD Simulation of Light Interaction with Cells
41
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